Hellooooo :D,
It’s been a very looong time of course :),
In this post we’ll talk about the meaning of a conjecture, and will give an example on it,
Of course our reference is the discrete text for Rosen,
Ok, let’s begin,
A conjecture is a statement that is being proposed to be a true statement, usually on the basis of some partial evidence, a heuristic argument, or the intuition of an expert.
When a proof of a conjecture is found, the conjecture becomes a theorem. Many times conjectures are shown to be false, so they are not theorems.
No matter how a conjecture was made, once it has been formulated, the goal is to prove or disprove it.
When mathematicians believe that a conjecture may be true, they try to find a proof. If they can’t find a proof, they may look for a counterexample(substitution of variables in the mathematical argument with some values that proves that the conjecture is not true). When they cannot find a counterexample, they may switch gears and once again try to prove the conjecture.
Although many conjectures are quickly settled, a few conjectures resist attack for hundreds of years and lead to the development of new parts of mathematics(What :D??!!!).
An example :),
Formulate a conjecture about the decimal digits that occur as the final digit of the square of an integer and prove your result.
Solution:
The smallest perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 255, and so on.
We notice that the digits that occur as the final digit of a square are 0, 1, 4, 5, 6, and 9, with 2, 3, 7, and 8 never appearing as the final digit of a square.
We conjecture this theorem: The final decimal digit of a perfect square is 0, 1, 4, 5, 6, or 9. How can we prove this theorem?
We first note that we can express an integer n as 10a + b(the bullet is shot from here :D), where a and b are positive integers and b is 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. Here a is the integer obtained by subtracting the final decimal digit of n from n and dividing by 10. Next, note that (10a + b)^2 = 100(a^2) + 20ab + b^2 = 10(10(a^2) + 2ab) + b^2, so that the final decimal digit of n^2 is the same as the final decimal digit of b^2.
Furthermore, note that the final decimal digit of b^2 is the same as the final decimal digit of (10 – b)^2(as the reader can verify ;)).
Consequently, we can reduce our poof to the consideration of six cases.
Case (1). The final digit of n is 1 or 9. Then the final decimal digit of n^2 is the final decimal digit of 1^2 = 1 or 9^2 = 81, namely, 1.
Case (2). The final digit of n is 2 or 8. Then the final decimal digit of n^2 is the final decimal digit of 2^2 = 4 or 8^2 = 64, namely, 4.
Case (3). The final digit of n is 3 or 7. Then the final decimal digit of n^2 is the final decimal digit of 3^2 = 9 or 7^2 = 49, namely, 9.
Case (4). The final digit of n is 4 or 6. Then the final decimal digit of n^2 is the final decimal digit of 4^2 = 16 or 6^2 = 36, namely, 6.
Case (5). The final decimal digit of n is 5. Then the final decimal digit of n^2 is the final decimal digit of 5^2 = 25, namely, 5.
Case (6). The final decimal digit of n is 0. Then the final decimal digit of n^2 is the final decimal digit of 0^2 = 0, namely, 0.
Because we have considered all six cases, we can conclude that the final decimal digit of n^2, where n is an integer is either 0, 1, 4, 5, 6, or 9.
I hope it was a useful post, if anything is not clear, just ask :)!
Impossible Means Nothing 